Question #ef7d5
2 Answers
The answer is the option
Explanation:
The parallelepiped volume is calculated by doing
Expanding
and
so finally
2)
Explanation:
We will use several facts.
First, note that the volume of a parallelepiped with coterminous edges
Two more facts about the scalar triple product we will use:
-
#vec(a) * (vec(b) xx vec(c)) = |(a_1, a_2, a_3), (b_1, b_2, b_3), (c_1, c_2, c_3)|# -
#vec(a) * (vec(b) xx vec(c)) = -vec(b) * (vec(a) xx vec(c)) = -vec(c) * (vec(b) xx vec(a))#
Note that the first fact means that any scalar triple product involving the same vector twice will clearly be
Finally, we will use that both the dot product and the cross product distribute over addition. With all that, we will now calculate the volume of the new parallelepiped as the scalar triple product of its coterminous edges:
#=|(vec(a)+vec(b)) * [(vec(a) xx vec(b)) + (vec(a) xx vec(c)) + (vec(c) xx vec(b)) + cancel((vec(c) xx vec(c))]|#
#=|cancel(vec(a) * (vec(a) xx vec(b))) + cancel(vec(a) * (vec(a) xx vec(c)))#
# + vec(a) * (vec(c) xx vec(b)) + cancel(vec(b) * (vec(a) xx vec(b)))#
# + vec(b) * (vec(a) xx vec(c)) + cancel(vec(b) * (vec(c) xx vec(b)))|#
#=|-vec(a) * (vec(b) xx vec(c)) - vec(a) * (vec(b) xx vec(c))|#
#=2|vec(a) * (vec(b) xx vec(c))|#
#=2(40)#
#=80#